What is the surface area of a square pyramid where the base edge is 8 ft. and the slant height is 16 ft.?(1 point) Responses 128 ft.2 128 ft. squared 320 ft.2 320 ft. squared 512 ft.2512 ft. squared 256 ft
To find the surface area of a square pyramid, we need to find the area of the base and the area of the four triangular faces.
The area of the base of the pyramid is calculated by finding the square of the length of one edge. In this case, the base edge is 8 ft, so the area of the base is 8^2 = 64 ft^2.
The area of each triangular face is found by using the formula A = 0.5bh, where b is the length of the base of the triangle (which is the same as the length of the edge of the pyramid's base) and h is the height of the triangular face. The slant height of the pyramid is given as 16 ft, so we can find the height of each triangular face using the Pythagorean theorem.
Using the formula a^2 + b^2 = c^2, where a and b are the sides of the right triangle (8 ft for the base and h for the height) and c is the hypotenuse (16 ft):
(8 ft)^2 + h^2 = (16 ft)^2
64 ft^2 + h^2 = 256 ft^2
h^2 = 256 ft^2 - 64 ft^2
h^2 = 192 ft^2
h = sqrt(192) ft ≈ 13.86 ft (rounded to two decimal places)
Therefore, the area of each triangular face is A = 0.5 * 8 ft * 13.86 ft ≈ 55.44 ft^2.
Since the square pyramid has four triangular faces, the total area of the four faces is 4 * 55.44 ft^2 = 221.76 ft^2.
Finally, we can find the total surface area by adding the area of the base and the area of the four triangular faces:
Total surface area = base area + 4 * triangular face area
Total surface area = 64 ft^2 + 221.76 ft^2
Total surface area ≈ 285.76 ft^2
Therefore, the surface area of the given square pyramid is approximately 285.76 ft^2.